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P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 57

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 57
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.1183826376515001962887748085765, 0.1183826376515001962887748085775]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.118382637

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 57 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "1.099964093",
      "0.124068294"
    ],
    [
      "0.863251293",
      "0.129052835"
    ],
    [
      "0.636401264",
      "0.196852982"
    ],
    [
      "1.477053801",
      "0.25860345"
    ],
    [
      "1.246026188",
      "0.31041115"
    ],
    [
      "0.435775185",
      "0.322578763"
    ],
    [
      "0.985924432",
      "0.331559863"
    ],
    [
      "0.754530548",
      "0.407619899"
    ],
    [
      "1.658633591",
      "0.414694728"
    ],
    [
      "1.406406779",
      "0.484583081"
    ],
    [
      "0.271274572",
      "0.503806063"
    ],
    [
      "1.175379166",
      "0.536390782"
    ],
    [
      "0.503604413",
      "0.549420099"
    ],
    [
      "0.94021402",
      "0.563870762"
    ],
    [
      "1.791795809",
      "0.612303508"
    ],
    [
      "0.718886735",
      "0.647966116"
    ],
    [
      "1.560768197",
      "0.664111208"
    ],
    [
      "0.165503669",
      "0.715632198"
    ],
    [
      "1.329740584",
      "0.715918908"
    ],
    [
      "1.096912547",
      "0.759829377"
    ],
    [
      "0.39783351",
      "0.761246234"
    ],
    [
      "0.870284156",
      "0.830000774"
    ],
    [
      "1.721148787",
      "0.83828314"
    ],
    [
      "0.612990909",
      "0.862344576"
    ],
    [
      "1.490121175",
      "0.89009084"
    ],
    [
      "1.259093562",
      "0.94189854"
    ],
    [
      "0.119919417",
      "0.947967885"
    ],
    [
      "0.352249259",
      "0.993581922"
    ],
    [
      "1.030664746",
      "1.004172705"
    ],
    [
      "1.881529378",
      "1.012455071"
    ],
    [
      "0.799637134",
      "1.055980405"
    ],
    [
      "1.650501766",
      "1.064262771"
    ],
    [
      "0.56613125",
      "1.095131365"
    ],
    [
      "1.419474153",
      "1.116070471"
    ],
    [
      "1.191045337",
      "1.178344637"
    ],
    [
      "0.137919249",
      "1.184366432"
    ],
    [
      "0.371245856",
      "1.229583881"
    ],
    [
      "0.960017725",
      "1.230152337"
    ],
    [
      "1.843835864",
      "1.252285962"
    ],
    [
      "0.720162069",
      "1.293467904"
    ],
    [
      "1.592709793",
      "1.293866524"
    ],
    [
      "1.361682181",
      "1.345674225"
    ],
    [
      "1.133253365",
      "1.40794839"
    ],
    [
      "0.220779512",
      "1.4123889"
    ],
    [
      "0.509093123",
      "1.422083041"
    ],
    [
      "0.902225752",
      "1.45975609"
    ],
    [
      "1.739969419",
      "1.47926447"
    ],
    [
      "1.508941807",
      "1.53107217"
    ],
    [
      "0.69319836",
      "1.570955662"
    ],
    [
      "1.27425207",
      "1.598151336"
    ],
    [
      "0.358626779",
      "1.60488806"
    ],
    [
      "1.043224458",
      "1.649959036"
    ],
    [
      "0.542732016",
      "1.753760681"
    ],
    [
      "1.44735342",
      "1.759686838"
    ],
    [
      "0.75981694",
      "1.848269575"
    ],
    [
      "1.229048929",
      "1.851343504"
    ],
    [
      "0.994224666",
      "1.881598446"
    ]
  ],
  "radius": "0.118382637"
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 57
}

起步布局答案

{
  "centers": [
    [
      "1.099964093",
      "0.124068294"
    ],
    [
      "0.863251293",
      "0.129052835"
    ],
    [
      "0.636401264",
      "0.196852982"
    ],
    [
      "1.477053801",
      "0.25860345"
    ],
    [
      "1.246026188",
      "0.31041115"
    ],
    [
      "0.435775185",
      "0.322578763"
    ],
    [
      "0.985924432",
      "0.331559863"
    ],
    [
      "0.754530548",
      "0.407619899"
    ],
    [
      "1.658633591",
      "0.414694728"
    ],
    [
      "1.406406779",
      "0.484583081"
    ],
    [
      "0.271274572",
      "0.503806063"
    ],
    [
      "1.175379166",
      "0.536390782"
    ],
    [
      "0.503604413",
      "0.549420099"
    ],
    [
      "0.94021402",
      "0.563870762"
    ],
    [
      "1.791795809",
      "0.612303508"
    ],
    [
      "0.718886735",
      "0.647966116"
    ],
    [
      "1.560768197",
      "0.664111208"
    ],
    [
      "0.165503669",
      "0.715632198"
    ],
    [
      "1.329740584",
      "0.715918908"
    ],
    [
      "1.096912547",
      "0.759829377"
    ],
    [
      "0.39783351",
      "0.761246234"
    ],
    [
      "0.870284156",
      "0.830000774"
    ],
    [
      "1.721148787",
      "0.83828314"
    ],
    [
      "0.612990909",
      "0.862344576"
    ],
    [
      "1.490121175",
      "0.89009084"
    ],
    [
      "1.259093562",
      "0.94189854"
    ],
    [
      "0.119919417",
      "0.947967885"
    ],
    [
      "0.352249259",
      "0.993581922"
    ],
    [
      "1.030664746",
      "1.004172705"
    ],
    [
      "1.881529378",
      "1.012455071"
    ],
    [
      "0.799637134",
      "1.055980405"
    ],
    [
      "1.650501766",
      "1.064262771"
    ],
    [
      "0.56613125",
      "1.095131365"
    ],
    [
      "1.419474153",
      "1.116070471"
    ],
    [
      "1.191045337",
      "1.178344637"
    ],
    [
      "0.137919249",
      "1.184366432"
    ],
    [
      "0.371245856",
      "1.229583881"
    ],
    [
      "0.960017725",
      "1.230152337"
    ],
    [
      "1.843835864",
      "1.252285962"
    ],
    [
      "0.720162069",
      "1.293467904"
    ],
    [
      "1.592709793",
      "1.293866524"
    ],
    [
      "1.361682181",
      "1.345674225"
    ],
    [
      "1.133253365",
      "1.40794839"
    ],
    [
      "0.220779512",
      "1.4123889"
    ],
    [
      "0.509093123",
      "1.422083041"
    ],
    [
      "0.902225752",
      "1.45975609"
    ],
    [
      "1.739969419",
      "1.47926447"
    ],
    [
      "1.508941807",
      "1.53107217"
    ],
    [
      "0.69319836",
      "1.570955662"
    ],
    [
      "1.27425207",
      "1.598151336"
    ],
    [
      "0.358626779",
      "1.60488806"
    ],
    [
      "1.043224458",
      "1.649959036"
    ],
    [
      "0.542732016",
      "1.753760681"
    ],
    [
      "1.44735342",
      "1.759686838"
    ],
    [
      "0.75981694",
      "1.848269575"
    ],
    [
      "1.229048929",
      "1.851343504"
    ],
    [
      "0.994224666",
      "1.881598446"
    ]
  ],
  "radius": "0.118382637"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

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